The equation provided in the image is:

$0.10x + 0.20y = 0.12(x + y)$

Problem breakdown:

• Lawrence is mixing two saline solutions:
  • $x$ milliliters of a 10% saline solution.
  • $y = 100$ milliliters of a 20% saline solution.
• The target is to create a 12% saline solution.

Step-by-Step Solution:

1. Substitute $y = 100$ into the equation: $0.10x + 0.20(100) = 0.12(x + 100)$ Simplify the terms: $0.10x + 20 = 0.12(x + 100)$

2. Distribute the 0.12 on the right-hand side: $0.10x + 20 = 0.12x + 12$

3. Move all terms involving $x$ to one side and constants to the other: $0.10x - 0.12x = 12 - 20$ Simplifying: $-0.02x = -8$

4. Solve for $x$: $x = \frac{-8}{-0.02} = 400$

Thus, Lawrence must use 400 milliliters of the 10% saline solution.

Final Answer:

Option C (400) is correct.

Would you like more details, or have any questions about this?

Here are some related questions:

1. What would happen if the final concentration was different, say 15% instead of 12%?
2. How would the formula change if Lawrence used a different amount of the 20% solution?
3. Can this type of equation be solved using matrix methods?
4. How can this problem be generalized for different concentrations of saline?
5. What if the solution concentrations weren’t given as percentages but as mass/volume?

Tip: When dealing with mixtures, it's often easier to first substitute known values before solving the equation!